**Augmenting the Great Cubicuboctahedron**

The great cubicuboctahedron, vertex figure (4,^{8}/_{3},3,^{8}/_{3})

Left: Each square face of the
great cubicuboctahedron can be augmented with an square
pyramid, the 4 in the vertex figure is replaced by a (3,3) The original vertex figures become
(__3,3__,^{8}/_{3},3,^{8}/_{3}) with new vertices of
(3,3,3,3). The original triangular faces are in blue, faces of the square
pyramids in yellow. All vertices lie on the convex hull.

Right: The above figure is shown with the square pyramids shown in framework style. A framework of this model is linked here.

Left: Another
augmentation of the great cubicuboctahedron has each face excavated. The
octagrammic faces are excavated with octagrammic
antiprisms, the square faces excavated with square
pyramids, and the triangular faces with triangular
antiprisms or octahedra. Each ^{8}/_{3} in the vertex figure is replaced by a
(3,3,3)
completing a cycle around the axis, each 4 in the vertex figure is replaced by a
(3,3)
completing a cycle around the axis and each 3 in the vertex figure is replaced by a
(3,3,3)
completing a cycle around the axis. The original vertex figures become (__3,3,3__,__3,3__,__3,3,3__,__3,3,3__)/3 - or
(3^{11})/3 with new vertices of (3,3,3,3,3,3,3,3)/3 - or (3^{8})/3, and
(3,3,3,3) from the octagrammic antiprisms, the square pyramids and octahedra
respectively. The triangles of the
octagrammic antiprisms are shown in purple, the triangles of the square pyramids are
blue and the octahedra are yellow. All
vertices are on the exterior of the figure.

Right: A
further
augmentation of the great cubicuboctahedron can be formed by excavating the
octagrammic faces with crossed octagrammic cupolas (or
8/5-cupolas, The cupolas themselves are not locally convex but the act of joining them to the
octagrams removes the retrograde polygon. The excavation is also shallow
enough not to cause the vertex figure to become crossed. Each ^{8}/_{3} in the vertex figure is replaced by a
(4,3,4)/2. The original vertex figures become (4,__4,3,4__,3,__4,3,4__)/3.
New vertices are added of the form (^{16}/_{5},3,4). This
augmentation is to date unique in that it contains polygons not found in any of
the uniform polyhedra. The ^{16}/_{5}-gons
are shown in pink. The square and triangular faces of the cupolas are red
and yellow respectively. All
vertices are on the exterior of the figure. _{ }